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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>All the results for the second order linear equations can be naturally extended to the <span class="process-math">\(n\)</span>-th order linear equations. Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_1.html">
\begin{equation}
\frac{\textrm{d}^n y}{\textrm{d} x^n}+P_1(x) \frac{\textrm{d}^{n-1} y}{\textrm{d} x^{n-1}}+\cdots+P_{n-1}(x) \frac{\textrm{d} y}{\textrm{d} x}+P_n(x) y=g(x).\tag{4.1.1}
\end{equation}
</div>
<p class="continuation">Introduce</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_1.html">
\begin{equation*}
L[y]=\frac{\textrm{d}^n y}{\textrm{d} x^n}+P_1(x) \frac{\textrm{d}^{n-1} y}{\textrm{d} x^{n-1}}+\cdots+P_{n-1}(x) \frac{\textrm{d} y}{\textrm{d} x}+P_n(x) y.
\end{equation*}
</div>
<p class="continuation">Then (<a href="" class="xref" data-knowl="./knowl/eq4_1.html" title="Equation 4.1.1">(4.1.1)</a>) can be rewritten as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_1.html">
\begin{equation}
L[y]=g(x).\tag{4.1.2}
\end{equation}
</div>
<p class="continuation">Initial conditions are proposed as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_1.html">
\begin{equation}
y(x_0)=b_0,\quad y^{\prime}(x_0)=b_1,\cdots, y^{(n-1)}(x_0)=b_{n-1}.\tag{4.1.3}
\end{equation}
</div>
<span class="incontext"><a href="sec4_1.html#p-145" class="internal">in-context</a></span>
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